Um, don't fall asleep? If you open your eyes, sure you will remember less of the dream, but isn't that better than not remembering any of it? I personally remember dreams prety well (eyes open or closed doesn't really matter to me) so long as I don't roll over or something. So try opening your eyes and remembering what you can, just don't pay any attention to what you see (sort of like when you day-dream) and you should be fine. Another thing you can try is to keep your eyes closed and sit up and put your head on your knees. This can be comfortable so you can focus on remembering your dreams, but it is rather difficult to fall asleep that way.

Well good luck mate.